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\title[ch05]{Chapter 20: AUTOMATIC PROOF OF IDENTITIES}
\author[]{SCC}
%\institute[XX大学]{XX大学\quad 数学与统计学院\quad 数学与应用数学专业}
%\date{2025年6月}

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% 封面页
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  \titlepage
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% 目录页
\begin{frame}{Contents}
  \tableofcontents
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% Section 1
\section{HOLONOMIC FUNCTIONS.}
%---------------------------------------------------
\begin{frame}{1.1 DEFINITION. }

A function is {\color{red}holonomic} if it is the solution of a holonomic module. 

Let $U$ be an open set of $\mathbb{R}^n$. If $f \in C^{\infty}(U)$, set

$$
\mathcal{I}_f = \{d \in A_n(\mathbb{R}) : d(f) = 0\}.
$$

Then $f$ is {\color{red}holonomic} if $A_n/\mathcal{I}_f$ is a holonomic module. 

But $A_n/\mathcal{I}_f$ is isomorphic to the submodule $A_nf$ of $C^{\infty}(U)$ generated by $f$. 

Thus $f$ is a holonomic function if and only if $A_nf$ is a holonomic module.

    
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%---------------------------------------------------
\begin{frame}{1.2 PROPOSITION. }

Let $f,g \in C^{\infty}(U)$ be holonomic functions. Then $f+g$ and $fg$ are holonomic.
    
\noindent\rule{\textwidth}{0.4pt}

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% Section 2
\section{HYPEREXPONENTIAL FUNCTIONS.}
%---------------------------------------------------
\begin{frame}{2.1 DEFINITION. }

A function $f \in C^{\infty}(U)$ is called {\color{red}hyperexponential} if $\partial_i(f)/f$ is a rational function of $x_1,\ldots,x_n$ for $i = 1,\ldots,n$. 

Note that if $q$ is a rational function in $\mathbb{R}(X)$, then $\exp(q)$ is hyperexponential. 

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%---------------------------------------------------
\begin{frame}{2.2 PROPOSITION. }

The product of hyperexponential functions is hyperexponential.
    
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%---------------------------------------------------
\begin{frame}{2.3 THEOREM. }
Suppose that $f$ is a hyperexponential function of $n$ variables. 

Then $\partial_i(f)/f = p_i/q_i$, where $p_i,q_i \in \mathbb{R}[X]$. 

Thus $f$ satisfies the system of differential equations
$
(q_i\partial_i - p_i)(f) = 0
$
for $1 \leq i \leq n$. 

Let $J = \sum_{i=1}^{n} A_n(q_i\partial_i - p_i)$. Then $J \subseteq \mathcal{I}_f$.

Put $M = A_n/J$ and let $q$ be the least common multiple of $q_1,\ldots,q_n$. 

Denote by $M[q^{-1}]$ the module $\mathbb{R}[X,q^{-1}] \otimes_{\mathbb{R}[x]} M$. 

Every element of $M[q^{-1}]$ can be written in the form $q^{-k} \otimes u$, where $u \in M$. 

Recall that the action of $\partial_i$ on $q^{-k} \otimes u \in M[q^{-1}]$ is defined by
$$
\partial_i(q^{-k} \otimes u) = -kq^{-k-1}\partial_i(q) \otimes u + q^{-k} \otimes \partial_iu.
$$

%\begin{theorem}
This theorem says that, if $M[q^{-1}]$ is holonomic, then $f$ is holonomic.
%\end{theorem}

    
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% Section 3
\section{THE METHOD.}
%---------------------------------------------------
\begin{frame}{3.1 STATEMENT OF THE PROBLEM, AND RESULT. }

Let $U = (a,b) \times (-\infty,\infty)\subset \mathbb{R}^2$ and $f$ be a function in $C^{\infty}(U)$ which satisfies
$$
\lim_{y \to \pm \infty} x^{\alpha}\partial_y^{\beta}f(x,y) = 0
$$
for all $\alpha,\beta \in \mathbb{N}^2$. 
%
Put
$$
R(x) = \int_{-\infty}^{+\infty} f(x,y)dy.
$$

Find a differential equation satisfied by $R(x)$.
    
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When $f$ is a holonomic function, there is an algorithm. 

\end{frame}

%---------------------------------------------------
\begin{frame}{3.2 FIRST STEP. }

Let $M = A_2/\mathcal{I}_f$ and $\pi : \mathbb{R}^2 \to \mathbb{R}$ be the projection on the first coordinate. The direct image of $M$ under $\pi$ is
$$
\pi_*M \cong M/\partial_yM \cong A_2/(\mathcal{I}_f + \partial_yA_2).
$$
If we now assume that $f$ is a holonomic function, then $M$ is a holonomic $A_2$-module. Thus $\pi_*M$ is holonomic over $A_1$. 

Hence the kernel of the homomorphism
$$
A_1 \to A_2/(\mathcal{I}_f + \partial_yA_2)
$$
which maps $d \in A_1$ to $d + (\mathcal{I}_f + \partial_yA_2)$ must be non-zero. 

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%---------------------------------------------------
\begin{frame}{3.3 EXAMPLE. }

Let $f(x,y) = \exp(-(x/y)^2 - y^2)$ and $R(x)$ be its integral between $-\infty$ and $\infty$. 

The function $f$ is hyperexponential, hence holonomic. 

A simple calculation shows that $f$ is a solution of the equations
$$
(y^2\partial_x + 2x)(f) = 0,\quad 
(y^3\partial_y + 2y^4 - 2x^2)(f) = 0.
$$

Let $L = \mathcal{I}_f + \partial_yA_2$... 

We get that
$$
D = 6\partial_x \cdot x + 8\partial_x \cdot x^2 + 8x - 2\partial_x^3 \cdot x^2
=
(x\partial_x + 3)(\partial_x - 2)(\partial_x + 2).
$$
satisfies the differential equation $D(R) = 0$. 

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% Section 4
\section{EXERCISES.}
%---------------------------------------------------
\begin{frame}{EXERCISE 1 }

Let $J_k$ be the left ideal of $A_n$ generated by the monomials $\partial^{\alpha}$ with $|\alpha| = k$. 

Show that $A_n/J_k$ is a holonomic $A_n$-module.

\noindent\rule{\textwidth}{0.4pt}

\textbf{Hint:} There exists an exact sequence,

$$
0 \to J_k/J_{k+1} \to A_n/J_{k+1} \to A_n/J_k \to 0
$$

and $J_k/J_{k+1} \cong K[X]$.

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%---------------------------------------------------
\begin{frame}{EXERCISE 2 }

Show that the derivative of a hyperexponential function is hyperexponential.


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%---------------------------------------------------
\begin{frame}{EXERCISE 3 }

Find an upper bound for the multiplicity of the module $A_n/\mathcal{I}_f$, when $f$ is a hyperexponential function.

%\textbf{Hint:} What is the bound on the multiplicity of $M[q^{-1}]$ in Theorem 2.5?


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%---------------------------------------------------
\begin{frame}{EXERCISE 4 }

Show that the function $\sin(xy)$ is holonomic, but not hyperexponential.


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%---------------------------------------------------
\begin{frame}{EXERCISE 5 }

Let $f(x,y) = \exp(-(x/y)^2 - y^2)$ and $R(x)$ be the integral of $f$ with respect to $y$ between $-\infty$ and $\infty$. Calculate $R(x)$ in two different ways, as follows:

\begin{enumerate}
    \item Using the substitution $t = y - x/y$ in $\int_{-\infty}^{\infty} \exp(-t^2)dt = (\sqrt{\pi})^{-1}$.
    \item Using differentiation under the integral sign to obtain the equation $R{\,}' = -2R$ and integrating it.
\end{enumerate}


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%---------------------------------------------------
\begin{frame}{EXERCISE 6 }

Let $f(x,y) = \exp(-xy^2)$ and $R(x) = \int_{-\infty}^{\infty} f(x,y)dy$. Find an operator $D \in A_1$ such that $D(R) = 0$.


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%---------------------------------------------------
\begin{frame}{EXERCISE 7 }

Let $\lambda_1,\ldots,\lambda_s \in \mathbb{R}^n$ and let $X$ be the $n$-tuple $(x_1,\ldots,x_n)$. Denote by $\lambda_i \cdot X$ the formal inner product of the two $n$-tuples. For $p_1,\ldots,p_s \in K[x_1,\ldots,x_n]$ put

$$
f(x) = \sum_{1}^{s} p_i(X)\exp(\lambda_i \cdot X).
$$

Show that $f$ is a holonomic function.


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